Page 253 - Mechanics of Microelectromechanical Systems

240 Chapter 4
layer, and the strains on it are an axial compressive strain due to the force
and a bending strain. Because the strains on this interface should be identical,
it follows that:

It should be noticed that the free strain is compressive (according to the
initial assumption), whereas the axial strain is extensional (the force has
the tendency of extending the top layer) as well as the bending strain (since
the interface fiber is under the neutral axis of the bent beam which has its
center of curvature upwards, as shown in Fig. 4.51 (b)). Similar reasoning
explains the signs of the strain components pertaining to the interface fiber of
the bottom layer – the right-hand side of Eq. (4.130).
Because there is no net axial force acting on the composite beam, it
follows that the two forces should be equal, namely:

As also indicated in Fig. 4.51 (b), there should be a relationship between the
bending effects produced on the right side section C-D of the form:

The bending moments and can be expressed according to the
engineering beam theory, as:

Equation (4.133) took into consideration that bending of the two layers takes
place independently, about the neutral (symmetry) axis of each component,
such that both deform as circles with the same curvature radius R. By
combining Eqs. (4.130) through (4.133), the unknown radius of curvature is
found to be:

Equation (4.134) is quite generic as the free strain can be generated by a
variety of means, for instance thermally, piezoelectrically or through shape-
memory effects. Each of these transduction solutions will be discussed
individually in the following.